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The Hausdorf dimension of the Apollonian packing of circles

P B Thomas and D Dhar

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We formulate the problem of determining the Hausdorf dimension, df, of the Apollonian packing of circles as an eigenvalue problem of a linear integral equation. We show that solving a finite-dimensional approximation to this infinite-order matrix equation and extrapolating the results provides a fast algorithm for obtaining high-precision numerical estimates for df. We find that df=1.305 686 729(10). This is consistent with the rigorously known bounds on df, and improves the precision of the existing estimate by three orders of magnitude.


PACS

02.30.Rz Integral equations

02.10.Ud Linear algebra

02.60.Nm Integral and integrodifferential equations

MSC

65R20 Integral equations

45C05 Eigenvalue problems (See also 34Lxx, 35Pxx, 45P05, 47A75)

45A05 Linear integral equations

Subjects

Mathematical physics

Computational physics

Dates

Issue 7 (7 April 1994)



  1. The Hausdorf dimension of the Apollonian packing of circles

    P B Thomas and D Dhar 1994 J. Phys. A: Math. Gen. 27 2257

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